Nyquist Meets Bode on the z-Plane

In this paper, we first present a new characterization of the Nyquist criterion in terms of Bode plots of the plant and the controller for discrete-time systems. This gives a nonparametric, and model independent characterization of arbitrary order stabilizing controllers. The result shows that the frequency response of any stabilizing controller must satisfy constraints on its magnitude, phase, and rate of change of phase at certain frequencies that are imposed by the frequency response of the inverse plant. We then show, by examples, that detailed and new results on controller design can be obtained for discrete time systems using these frequency domain criteria.